REPORT ON PHYSICAL OPTICS. 389 



chy in his able researches on the propagation of waves iii elastic 

 media, although he does not seem to have been aware of all its 

 properties. The properties of the same surface, and its use in 

 constructing the direction of a reflected or a refracted ray, were 

 also discovered, independently, by Mr. M'CuUagh, who has re- 

 cently applied them to the geometrical development of the theory 

 of double refraction*. 



The relations between the surface of wave-slowness and that 

 of the wave have led Professor Hamilton to the discovery of 

 some new geometrical properties of the latter. These properties 

 are demonstrated by means of certain transformations of the 

 equation of the wave-surface ; and it is shown that this surface 

 has four conoidal cusps, at the extremities of the lines of single 

 ray-velocity, at each of which the wave is touched, not by two 

 planes as Fresnel supposed, but by an infinite number forming 

 a tangent cone of the second degree ; while, at the extremities 

 of the lines oi single tvave-velocity, there are jfour circles of plane 

 .contact, in every point of each of which the wave-surface is 

 touched by a single plane. These singiilar properties have led 

 Professor Hamilton to anticipate two new laws of refraction — 

 called by him conical refraction, because in each case a single 

 ray is refi-acted into an infinite number forming a species of cone. 

 External conical refraction corresponds to the cusp on the wave- 

 surface ; and takes place without, when a single internal ray 

 coincides with either of the lines of single ray- velocity. Internal 

 conical refraction, on the other hand, takes place within the 

 crystal, when a single ray is incident externally at an angle cor- 

 responding to the line of single wave-velocity within. In this 

 latter case, if the crystal be bounded by parallel planes, all the 

 rays of the cone will emerge at the second surface parallel to the 

 ray incident on the first, so as to form a small elliptic cylinder, 

 whose magnitude will depend upon the angle of the cone and 

 the thickness of the crystal. All these remarkable conclusions 

 have been verified in the fullest manner by experiment f. 



I shall now proceed to give a brief account of the labours 

 of M. Cauchy in this interesting department of analysis. The 

 researches of this eminent mathematician, on the propaga- 

 tion of motion in elastic media, are scattered through various 

 livraisons of the Efxercices de Mathematiques ', and he has 

 given a valuable summary of the results of these investigations, 



• " Geometrical Propositions applied to the Wave-theory of Light," Trans- 

 actions of the Royal Irish Academy, vol. xvii. 



f " On the Phenomena presented by Light in its passage along the axes of 

 biaxal Crystals," Ibid. 



